Lightlike radial null geodesic - how do we know it has constant theta and phi?

[andrewkirk]

Consider a light ray emanating from the origin of a FLRW coordinate system in a homogeneous, isotropic universe. The initial velocity of that ray will have only x⁰ (t) and x¹ (r) components. In papers I have seen it is assumed that its velocity will continue to have zero circumferential components: x² (θ) and x³ ($\phi$), in other words that θ and $\phi$ are constant.

A loose argument for this is that, for the geodesic to develop any circumferential components would identify a preferred direction in space, thereby contradicting the isotropy assumption. I find this unconvincing, as the 'direction' is a coordinate-dependent artifact, and hence does not necessarily have any physical significance. The isotropy assumption is a coordinate-independent statement about the nature of the spacetime, not about a particular coordinate system (well, perhaps it does contain information about the time coordinate, as it seems to state that the constant-time hypersurfaces are isotropic, but those hypersurfaces can be parameterised in an infinity of different ways, so there's nothing significant about a particular spherical choice of coordinates as in the FLRW system.).

Is there a more rigorous argument as to why the null geodesic cannot have any circumferential components?

 

[fzero]

The isotropy argument is perfectly physical, but this also follows from the geodesic equation, since

$$\Gamma^\theta_{rr}=\Gamma^\theta_{rt}=\Gamma^\theta_{tt}=0$$

and similarly for $\phi$. I think your concerns about other coordinate systems is just explained by noting that the spherical symmetry would look different in arbitrary coordinates

 

[andrewkirk]

this also follows from the geodesic equation, since
$$\Gamma^\theta_{rr}=\ \Gamma^\theta_{rt}=\ \Gamma^\theta_{tt}=0$$
and similarly for $\phi$.

Geodesic Equation:
$\frac{d^2x^\theta}{d\lambda^2}\ +\ \Gamma^\theta_{\alpha\beta}\ \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda}$

Thank you for your reply fzero. I can see that those zero values of the Christoffel symbols are correct and at first I thought that was enough to guarantee the result, via the geodesic equation, but on reflection, I think that alone is not sufficient to give the result.

Although the zero values of $\Gamma^\theta_{rr}\text{, }\Gamma^\theta_{rt}\text{, }\Gamma^\theta_{tt}$ ensure that those terms of the geodesic equation are zero everywhere, there are plenty of other Christoffel symbols in that equation that are nonzero, such as $\Gamma^\theta_{\theta r}$.

At first I thought we could disregard terms like that because $\frac{dx^\theta}{d\lambda}(0)=\frac{d^2x^\theta}{d\lambda^2}(0)=0$. I now see that that is insufficient argument. There are plenty of functions for which the first and second derivatives at a point are zero but which subsequently become nonzero, for example $y=x^3\text{ at }x=0$.

What arguments can be employed together with the above observation about the Christoffel symbols to reach a conclusion that $\frac{dx^\theta}{d\lambda}=0$ everywhere along the radial geodesic? Perhaps using some additional info from the metric?

 

[andrewkirk]

I have managed to complete the proof. The attached file is a TeX formatted version of the proof, together with a corollary that a vector with no circumferential components that is parallel transported along the radial curve does not gain any circumferential components.

The proof relied heavily on the observations by fzero that $\Gamma^\theta_{tt}= \Gamma^\theta_{rt} = \Gamma^\theta_{rr}=0$, but also had to use some nonzero values of other Christoffel symbols, which turned out to lead to key cancellations.

 

[sardar]

The assumption that the lightlike radial null geodesic has constant theta and phi is based on the fundamental principles of homogeneous and isotropic universes. These principles state that the universe looks the same at every point and in every direction, and this is reflected in the FLRW coordinate system. Since the universe is assumed to be isotropic, there is no preferred direction or axis, and therefore the geodesic must remain constant in theta and phi.

Additionally, the isotropy assumption is not just a coordinate-dependent artifact, but a fundamental principle of the universe. It is a key feature of the cosmological principle, which states that the universe is both homogeneous and isotropic on large scales. This principle has been supported by numerous observations, including the cosmic microwave background radiation, which shows a nearly uniform temperature in all directions.

Furthermore, the assumption of constant theta and phi is also supported by the fundamental laws of physics, specifically the principle of conservation of angular momentum. Since the light ray is traveling in a straight line, it cannot have any circumferential components as this would violate the law of conservation of angular momentum.

In conclusion, the assumption of constant theta and phi for the lightlike radial null geodesic is not just a coordinate choice, but is based on the fundamental principles of the universe and the laws of physics. It is a necessary assumption in order to maintain the homogeneity and isotropy of the universe and is supported by observational evidence. Therefore, there is a rigorous argument for why the null geodesic cannot have any circumferential components.